Add outer_approximation_by_rectangles

include/shape/rectangular_approximation.hpp

@@ -0,0 +1,70 @@
+#pragma once
+
+#include "shape/shape.hpp"
+
+namespace shape
+{
+
+struct OuterApproximationByRectanglesInputShape
+{
+    /** Shape to approximate. */
+    ShapeWithHoles shape;
+
+    /**
+     * Number of copies of the shape: the waste of its rectangles is counted
+     * once for each copy.
+     */
+    ShapePos copies = 1;
+};
+
+/**
+ * Approximate each shape of 'shapes' by axis-aligned rectangles whose union
+ * contains it.
+ *
+ * The waste of a shape is the area of its rectangles minus the area of the
+ * shape. The waste of all the shapes, counted once for each copy, is at most
+ * 'maximum_waste', and, within it, the number of rectangles is reduced as
+ * much as possible.
+ *
+ * The circular arcs of each shape are first replaced by line segments
+ * outside of them (see approximate_by_line_segments); then, the shape is
+ * decomposed into trapezoids (see trapezoidation), and each trapezoid is
+ * covered by its bounding box. A shape whose edges are all horizontal or
+ * vertical, such as a rectangular frame, is thus covered exactly. Then:
+ * - as long as the waste is larger than 'maximum_waste', the rectangle of a
+ *   single trapezoid whose split into two rectangles of half its height
+ *   reduces the waste the most is split, as long as its shape has fewer than
+ *   'maximum_number_of_rectangles' rectangles;
+ * - then, as long as the waste stays at most 'maximum_waste', the two
+ *   rectangles of a same shape whose replacement by their bounding box
+ *   increases the waste the least are merged.
+ * Both choices are made across all the shapes, weighted by their copies.
+ *
+ * Where rectangles of a shape overlap, the area of the overlap counts
+ * several times in their total area: the waste of the union of the
+ * rectangles is at most the waste as defined above.
+ *
+ * If the waste can't be reduced to 'maximum_waste' within
+ * 'maximum_number_of_rectangles' rectangles per shape (for example, with
+ * 'maximum_waste' equal to 0 and a shape with an oblique edge), the waste
+ * is larger than 'maximum_waste'.
+ *
+ * Return, for each shape, its rectangles.
+ */
+std::vector<std::vector<AxisAlignedBoundingBox>> outer_approximation_by_rectangles(
+        const std::vector<OuterApproximationByRectanglesInputShape>& shapes,
+        AreaDbl maximum_waste,
+        ElementPos maximum_number_of_rectangles = 256);
+
+/**
+ * Write the inputs of outer_approximation_by_rectangles to a JSON file in
+ * the format of the test fixtures in
+ * data/tests/rectangular_approximation/outer_approximation_by_rectangles/.
+ */
+void outer_approximation_by_rectangles_export_inputs(
+        const std::string& file_path,
+        const std::vector<OuterApproximationByRectanglesInputShape>& shapes,
+        AreaDbl maximum_waste,
+        ElementPos maximum_number_of_rectangles = 256);
+
+}

src/CMakeLists.txt

@@ -15,6 +15,7 @@ target_sources(Shape_shape PRIVATE
     offset.cpp
     simplification.cpp
     approximation.cpp
+    rectangular_approximation.cpp
     supports.cpp
     rasterization.cpp
     no_fit_polygon.cpp